On automorphism groups of global function fields

Abstract

In this thesis, we first give a characterization of fixed fields under subgroups of the decomposition group $\mathcal{A}(P_\infty)$ of the Deligne--Lusztig function fields. More specifically, we give a characterization of subgroups in the decomposition group by means of a necessary and sufficient condition. By establishing an analogue of Kneser's theorem on the Hermitian product over vector spaces, we determine the genera set consisting of all the genera of fixed fields of subgroups of the decomposition group for the Hermitian function field with an odd characteristic. In addition, we also improve the results for other cases of the Deligne--Lusztig function fields. In the second part, we concentrate on determining the automorphism groups of cyclotomic function fields with modulus $x^{n+1}$ and $P$ over the rational function fields, where $n\ge 1$ and $P$ is an irreducible polynomial of degree two. We also investigate the automorphism groups of some subfields of cyclotomic function fields.